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Although I haven't starting implementing training yet, I found an excellent training algorithm that will mesh really well with my app's design. It's called Cascade-Correlation (pdf). Basically, it adds one node at a time to the system, and trains a node as soon as it's added. Once the node is in place, it's frozen and isn't trained any more. So you can get higher accuracy by just continuing to add nodes. Apparently the whole thing works faster (fewer epochs) than back-prop, because training one node at a time solves the moving-target problem.

This algorithm is a good fit for two reasons:
1) my program keeps track of individual nodes seperately anyway (most neural networks and algorithms work with entire layers rather than individual nodes).

2) there's going to be both user-created nodes and training-created nodes in the system, and the last thing the user wants is for the training algorithm to f with the nodes that he put there by hand. So I need algorithms that can work without changing the net that's already there.

And as a side note, that paper is yet *another* connectionist-related development to come out of CMU. I'm looking at places to go for grad school, and the CNBC is definitely first on my list.
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